Question: What is the positive difference between the sum of the first 20 positive even integers and the sum of the first 15 positive odd integers?
The sum of the first 20 positive even integers is $2 + 4 + \dots + 40 = 2(1 + 2 + \dots + 20)$.  For all $n$, $1 + 2 + \dots + n = n(n + 1)/2$, so $2(1 + 2 + \dots + 20) = 20 \cdot 21 = 420$.

The sum of the first 15 positive odd integers is $1 + 3 + \dots + 29$.  The sum of an arithmetic series is equal to the average of the first and last term, multiplied by the number of terms, so this sum is equal to $(1 + 29)/2 \cdot 15 = 225$.  The positive difference between these sums is $420 - 225 = \boxed{195}$.